On the axiomatizability of $\mathrm{C}^*$-algebras as operator systems

TL;DR

This paper demonstrates that unital C*-algebras can be characterized as an elementary class within the language of operator systems, providing a logical framework for their axiomatization and defining their multiplication.

Contribution

It establishes the elementary class status of unital C*-algebras in operator systems and identifies the precise logical complexity of their axiomatization.

Findings

Unital C*-algebras form an elementary class in operator systems.

A definable predicate characterizes multiplication in C*-algebras.

The class is $orall hereforeorall$-axiomatizable but not simpler.

Abstract

We show that the class of unital $\mathrm{C}^*$-algebras is an elementary class in the language of operator systems. As a result, we have that there is a definable predicate in the language of operator systems that defines the multiplication in any $\mathrm{C}^*$-algebra. Moreover, we prove that the aforementioned class is $\forall\exists\forall$-axiomatizable but not $\forall\exists$-axiomatizable nor $\exists\forall$-axiomatizable.